Work in time-domain astronomy necessitates robust, automated data processing pipelines that operate in real time. We present the BANZAI pipeline which processes the thousands of science images produced across the Las Cumbres Observatory Global Telescope (LCOGT) network of robotic telescopes each night. BANZAI is designed to perform near real-time preview and end-of-night final processing for four types of optical CCD imagers on the three LCOGT telescope classes. It performs instrumental signature removal (bad pixel masking, bias and dark removal, flat-field correction), astrometric fitting and source catalog extraction. We discuss the design considerations for BANZAI, including testing, performance, and extensibility. BANZAI is integrated into the observatory infrastructure and fulfills two critical functions: (1) real-time data processing that delivers data to users quickly and (2) derive metrics from those data products to monitor the health of the telescope network. In the era of time-domain astronomy, to get from these observations to scientific results, we must be able to automatically reduce data with minimal human interaction, but still have insight into the data stream for quality control.